After our Jelly Belly factory tour we drove to Madison, WI to spend the night. The next morning, we drove to Frank Lloyd Wright’s Taliesin so my wife could tour the house and grounds (she loved it) and then drove a few more hours into Minnesota to stay for the night. It was a relatively low-activity day for us, but it set up the days to follow and gave me plenty of time to think.
I keep coming back to the “just right” questions I mentioned the other day. I’m looking for things around us that our students can observe and wonder about, so that these questions can then be used to lead us to theoretical math. This may well be possible, and a comment by my colleague Bill Enos on my earlier post helped me to see where these just right questions might be found. However, in all my looking around, I forgot about one crucial thing: not all questions in math are based on real-world observations, nor should they be.
One of the greatest things about math is that it exists independent of the world that describes it. The fact that so much of what we observe can be described mathematically is truly wonderful, but even if that weren’t true math would still exist and still be just as great. This is one of the things that separates math from the other disciplines, but it also makes it challenging to teach at times.
I tell my students to think like scientists, but without observing anything more than the math in front of them. A biologist might look at a population of bateria and wonder how they might change if she decreases the amount of light they get. Similarly, a mathematician might look at a relationship and wonder what would happen if some underlying assumption isn’t true. My goal for my students is for them to understand this and to begin to ask these questions themselves.
These are pure math questions, very different than the kinds of questions that might arise from watching a wind turbine rotate, and the kinds of answers they can get are just as different as the questions. So really, I have two challenges: help my students learn to ask the just right questions about the world around them, and help them learn to ask just right theoretical questions about the math in front of them. I’m wondering if we can help our younger students learn the former so that they can transition to the latter in the following years.
There’s a lot of conversation in education about higher-order questioning skills, but it all seems to be focused on the world around us. What I’m asking feels different because we have to leave the observable world and bring our questioning into the theoretical world. It looks like it’s time for me to loook for some research on these topics. If anyone has suggestions for things I should read, I’d love to hear (them).